On the description of \(\mathbb Z/p^2\mathbb Z\)-torsors by the Kummer-Artin-Schreier-Witt theory
type: Item
instance of: scholarly article
title: On the description of \(\mathbb Z/p^2\mathbb Z\)-torsors by the Kummer-Artin-Schreier-Witt theory
zbMATH Open document ID: 1085.14038
author: Kazuyoshi Tsuchiya
published in: Tokyo Journal of Mathematics
publication date: 2004-05-18T00:00:00Z
review text: Let \(p\) be a prime and let \(\zeta _{n}\) be a primitive \(p^{n}\)-th root of unity. For \(A=\mathbb Z_{\left( p\right) }\left[ \zeta _{n}\right] \) we have the Kummer-Artin-Schreier-Witt sequence \[ 0\rightarrow \left(\mathbb Z/p^{n} \mathbb Z\right) _{A}\rightarrow \mathcal{W}_{n}\rightarrow \mathcal{V}_{n}\rightarrow 0 \] where \(\mathcal{W}_{n}\) and \(\mathcal{V}_{n}\) are certain group schemes related to deformations of \(\mathbf{G}_{a}\) and \(\mathbf{G}_{m}\). This sequence gets its name from the fact that its special fiber is the Artin-Schreier-Witt sequence and its generic fiber is the Kummer sequence. Given a scheme \(X\) and a flat group scheme \(G\) which is locally of finite type over \(X\) one can consider the set PHS\(\left( G/X\right) \) of isomorphism classes of \(G\)-torsors over \(X\). This paper examines the case where \(G= \mathbb Z/p^{2}\mathbb Z\) and \(X\) is an \(A\)-scheme. In other words, a description of the unramified cyclic coverings of degree \(p^{2}\) over \(X\). The main result is as follows. Let \(\mathcal{U=}\left\{ U_{j}\right\} \) be an affine open covering on \(X\), and let \(\Psi ^{2}:\mathcal{W} _{2}\rightarrow \mathcal{V}_{2}\) be the map in the Kummer-Artin-Schreier-Witt sequence in the case where \(n=2\). (This is the only case in which the author explicitly describes these group schemes and this map, however it is also the only case used in this paper.) Let \(\mathbf{ f}_{ij}\in Z^{1}\left( \mathcal{U},\mathcal{W}_{2}\right) \) be such that \( \Psi ^{2}\left( [\mathbf{f}_{ij}]\right) =0.\) Then for each \(j\) there exists a \(\mathbf{b}_{j}\in \Gamma \left( U_{j},\mathcal{V}_{2}\right) \) such that \( \Psi ^{2}\left( \mathbf{f}_{ij}\right) =\left( \Lambda _{0}^{G}\left( \mathbf{b}_{j},I^{G}\left( \mathbf{b}_{i}\right) \right) ,\Lambda _{1}^{G}\left( \mathbf{b}_{j},I^{G}\left( \mathbf{b}_{i}\right) \right) \right) \) on \(U_{j}\cap U_{i}.\) For \(\mathbf{h\in \Gamma }\left( X,\mathcal{V }_{2}\right) \) a \(\mathbb Z/p^{2…
reviewed by: Alan Koch
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 14F20
zbMATH DE Number: 2064100
zbMATH Keywords: group schemes
zbMATH Keywords: torsors
zbMATH Keywords: Kummer Theory
zbMATH Keywords: Artin-Schreier Witt Theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208687
OpenAlex ID: W2051140576
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DOI: 10.3836/TJM/1244208687
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description: scientific article; zbMATH DE number 2064100